Question

An open box of maximum volume is to be made from a square piece of tin sheet 24cm on a side by cutting equal squares from the corners and turning of the sides.

(i) Complete the following table.

Height of the box (x.cm)	Width of the box	Volume of the box (v. cm3)
1	24 - 2 x 1	1 x (24 - 2 x 1)2 = 484
2	24 - 2 x 2	2 x (24 -2 x 2)2 = 800
3	.......	........
4	........	........
5	........	........
6	........	........

(ii) Using the above table, express V as a function of x and determine its domain.

(iii) Find height (x. cm) of the box when volume V is maximum by differentiation. 

Answer 1

Height of the box (x.cm)	Width of the box	Volume of the box (v. cm3)
1	24 - 2 x 1	1 x (24 - 2 x 1)2 = 484
2	24 - 2 x 2	2 x (24 -2 x 2)2 = 800
3	24 - 2 x 3	3 x (24 - 2 x 3)2 = 972
4	24 - 2 x 4	4 x (24 - 2 x 4)2 = 1024
5	24 - 2 x 5	5 x (24 - 2 x 5)2 = 980
6	24 - 2 x 6	6 x (24 - 2 x 6)2 = 864

(ii) Generalise the above table as a function.

V = x(24-2x)², 0 < x < 12.

(iii) \(\frac{dV}{dx}\) = x.2(24 – 2x)(-2) + (24 – 2x)²

= -4x(24 – 2x) + (24 – 2x)²

= -96x + 8x² + 576 + 4x² – 96x

= 12x² – 192x + 576

For maximum or minimum,

Therefore volume is maximum when x = 4 cm.